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Chapter 1: Problem 8

Use inductive reasoning to determine the next element in each list. $$1,3,4,7,11,18,29$$

Short Answer

Expert verified
The next element is 47.

Step by step solution

01

- Identify the pattern

Observe the differences between consecutive elements: \( 3 - 1 = 2 \), \( 4 - 3 = 1 \), \( 7 - 4 = 3 \), \( 11 - 7 = 4 \), \( 18 - 11 = 7 \), \( 29 - 18 = 11 \).
02

- Analyze the differences

The differences are 2, 1, 3, 4, 7, and 11. Check the pattern of these differences: the differences themselves form the sequence \( 2, 1, 3, 4, 7, 11 \).
03

- Identify the next difference

The sequence of differences can be analyzed further. Notice that 3, 4, 7, and 11 look like they follow a pattern where the difference sequential sums: \( 1+2=3 \), \( 3+1=4 \), \( 7=3+4 \), \( 11=4+7 \). Therefore, the next difference should be \( 7+11=18 \).
04

- Calculate the next element

Add the next difference, 18, to the last given element, which is 29. So, \( 29 + 18 = 47 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number Patterns
Observing number patterns is like finding hidden codes in numbers. When faced with sequences, the first thing to do is look at how the numbers evolve. For instance, in our given sequence \(1, 3, 4, 7, 11, 18, 29\), the discrepancies between these numbers tell us there's a pattern to find. Analyzing how each number progresses lets you predict the next number accurately. This pattern recognition is crucial for making sense of the sequence.
Sequence Analysis
Sequence analysis involves examining each element and its relationship with the others. Let's look at our sequence: \(1, 3, 4, 7, 11, 18, 29\). By determining the differences between each consecutive number, we see:
  • \(3 - 1 = 2\)
  • \(4 - 3 = 1\)
  • \(7 - 4 = 3\)
  • \(11 - 7 = 4\)
  • \(18 - 11 = 7\)
  • \(29 - 18 = 11\)
These form another sequence: \(2, 1, 3, 4, 7, 11\). By analyzing this sequence of differences, we notice that each number is formed by adding the previous two: \(2, 1\), then \(1 + 2 = 3\), \(3 + 1 = 4\), \(4 + 3 = 7\), \(7 + 4 = 11\). Recognizing these patterns helps us understand the structure of the original numbers.
Mathematical Differences
Mathematical differences are the gaps between successive numbers in a sequence. We work with these differences to find new patterns. Take our example: the differences form \(2, 1, 3, 4, 7, 11\). This sequence of differences grows by adding the previous numbers together. To predict the next difference, add the last two: \(11 + 7 = 18\). To find the sequence's next number, add this difference to the last number: \(29 + 18 = 47\). So, the next element in the sequence is \(47\). Understanding these differences and how they evolve is key to mastering sequence analysis.

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